Ouch! I think it's just a modulus function, that's what the graph looks like at least: Show[ Plot[ArcTan[Cos[x], Sin[x]], {x, -10 Pi, 10 Pi}, PlotStyle -> Thick], Plot[Mod[x + Pi, 2 Pi] - Pi, {x, -10 Pi, 10 Pi}, PlotStyle -> Directive[Dashing[.05], Thick, Red]], PlotRange -> All, ImageSize -> 500 ] Or is it supposed to be: Show[ Plot[ArcTan[Cos[x], Sin[x]], {x, -10 Pi, 10 Pi}, PlotStyle -> Thick], Plot[Pi - Mod[Pi - x, 2 Pi], {x, -10 Pi, 10 Pi}, PlotStyle -> Directive[Dashing[.05], Thick, Red]], PlotRange -> All, ImageSize -> 500] Gotta watch those boundary values, and don't forget all of the broken evaluation over complexes: ArcTan[Cos[x],Sin[x]]==(Pi-Mod[Pi-Re[x],2*Pi]+I*Im[x]) LHS Sometimes returns precision errors? What the bleep is it doing numerics for an identity map? Try: ArcTan[Cos[I #], Sin[I #]] & /@ Range[100] ArcTan[Sin[I #]/Cos[I #]] & /@ Range[100] Refine[ArcTanh[Tanh[x]], {x \[Element] Reals}] Surprisingly, the second alternative does quite a bit better, managing to get up to n=75 rather than n=18, but still dies at 76 and thereafter. This is okay as long as we only need a thin strip around the real axis, but in general, no, "Indeterminate" is not a good return. --Brad